Fourier method for the Neumann problem on a torus

Abstract The Fourier method approach to the Neumann problem for Laplacian operator in case of a solid torus contrasts many respects with much more straight forward situation ball 3-space. Although Dirichlet-to-Neumann map can be readily expressed terms series expansions toroidal harmonics, we show that resulting equations contain undetermined parameters which cannot calculated algebraically. A rapidly computing numerical solutions is presented illustrations. results interior and exterior domains combine provide solution shell between two tori.